# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021013
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## Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays

 School of Mathematics, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Weihua Jiang

Received  October 2020 Revised  November 2020 Early access December 2020

Fund Project: The authors are supported by the National Natural Science Foundation of China (No. 11871176)

We consider a two-species Lotka-Volterra competition system with both local and nonlocal intraspecific and interspecific competitions under the homogeneous Neumann condition. Firstly, we obtain conditions for the existence of Hopf, Turing, Turing-Hopf bifurcations and the necessary and sufficient condition that Turing instability occurs in the weak competition case, and find that the strength of nonlocal intraspecific competitions is the key factor for the stability of coexistence equilibrium. Secondly, we derive explicit formulas of normal forms up to order 3 by applying center manifold theory and normal form method, in which we show the difference compared with system without nonlocal terms in calculating coefficients of normal forms. Thirdly, the existence of complex spatiotemporal phenomena, such as the spatial homogeneous periodic orbit, a pair of stable spatial inhomogeneous steady states and a pair of stable spatial inhomogeneous periodic orbits, is rigorously proved by analyzing the amplitude equations. It is shown that suitably strong nonlocal intraspecific competitions and nonlocal delays can result in various coexistence states for the competition system in the weak competition case. Lastly, these complex spatiotemporal patterns are presented in the numerical results.

Citation: Xianyong Chen, Weihua Jiang. Multiple spatiotemporal coexistence states and Turing-Hopf bifurcation in a Lotka-Volterra competition system with nonlocal delays. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021013
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##### References:
$w_{*}>0$ is a positive root of $F(w) = 0$
shows. The left figure represents the enlarged figure inside the pink rectangular part of the right figure and the red hollow circles are intersection points of $\mathcal{L}_{n}$ and $\mathcal{S}_{n}$. The solid black circles means that the remaining curves $\mathcal{L}_{n}$ and $\mathcal{S}_{n}$ are omitted here">Figure 2.  System parameter values are taken as Table 1 shows. The left figure represents the enlarged figure inside the pink rectangular part of the right figure and the red hollow circles are intersection points of $\mathcal{L}_{n}$ and $\mathcal{S}_{n}$. The solid black circles means that the remaining curves $\mathcal{L}_{n}$ and $\mathcal{S}_{n}$ are omitted here
(a), (b) are bifurcation sets and phase portraits respectively
When the initial values $u_{1}(t,x) = 0.45-0.01$, $u_{2}(t,x) = 0.27-0.01$, $t\in[-\tau,0]$ and parameters $(\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(-0.01,0.01)\in \mathscr{D}_{1}$, the coexistence equilibrium is stable
When the initial values $u_{1}(t,x) = 0.45-0.01$, $u_{2}(t,x) = 0.27-0.01$, $t\in[-\tau,0]$ and parameters $(\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(0.01,0.01)\in \mathscr{D}_{2}$, a spatial homogeneous periodic orbit is stable
For $(\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(0.003,-0.01)\in \mathscr{D}_{4}$, (a), (b) with initial values $u_{1}(t,x) = 0.45-0.01\cos x$, $u_{2}(t,x) = 0.27-0.01\cos x$ and (c), (d) with initial values $u_{1}(t,x) = 0.45+0.01\cos x$, $u_{2}(t,x) = 0.27+0.01\cos x$, , $t\in[-\tau,0]$. A pair of spatial inhomogeneous periodic orbits is stable, which indicates $\mathscr{D}_{4}$ is a bistable region
For $(\tau,d_{1}) = (\tau_{w_{*}},d_{1}^{1}(d_{2}))+(-0.01,-0.01)\in \mathscr{D}_{6}$, (a), (b) with initial values $u_{1}(t,x) = 0.45-0.01\cos x$, $u_{2}(t,x) = 0.27-0.01\cos x$ and (c), (d) with initial values $u_{1}(t,x) = 0.45+0.01\cos x$, $u_{2}(t,x) = 0.27+0.01\cos x$, , $t\in[-\tau,0]$. a pair of spatial inhomogeneous steady states is stable, which indicates $\mathscr{D}_{6}$ is a bistable region
Values of system parameters
 Parameters $r_{1}$ $r_{2}$ $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $b_{11}$ $b_{12}$ $b_{21}$ $b_{22}$ Values $5$ $5$ $4$ $7$ $6$ $5$ $3$ $1$ $0.5$ $4$
 Parameters $r_{1}$ $r_{2}$ $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $b_{11}$ $b_{12}$ $b_{21}$ $b_{22}$ Values $5$ $5$ $4$ $7$ $6$ $5$ $3$ $1$ $0.5$ $4$
Values of system parameters
 Parameters $d_{2}$ $r_{1}$ $r_{2}$ $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $b_{11}$ $b_{12}$ $b_{21}$ $b_{22}$ Values $0.6$ $5$ $5$ $4$ $7$ $6$ $5$ $3$ $1$ $0.5$ $4$
 Parameters $d_{2}$ $r_{1}$ $r_{2}$ $a_{11}$ $a_{12}$ $a_{21}$ $a_{22}$ $b_{11}$ $b_{12}$ $b_{21}$ $b_{22}$ Values $0.6$ $5$ $5$ $4$ $7$ $6$ $5$ $3$ $1$ $0.5$ $4$
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